Starting from the Greeks Leucippus and Democritus, plenty of western philosophers and natural scientists alike have subscribed to a deterministic universe. Indeed, the precise laws of Newtonian mechanics have always instilled a confidence in 18th and 19th century physicists that the world at large can not only be described, but be predicted with clockwork accuracy given sufficient knowledge of the initial condition. Laplace famously in 1814 described his demon: suppose an entity exists such that it knows the precise location and momentum of every atom in the universe, then it can predict, using Newton's laws, the evolution of events. Using Hamilton's theory of dynamics, we see that the Newtonian/Laplacian world view is clearly embodied in the fact that a universe described purely by interactions in accordance to the three Newton laws--inertia; relation of force, mass, and acceleration; and symmetry of action--is described by a system of integrable ordinary differential equations. So mathematically the determinism world view is sound, at least on such a minimalistic universe.

The advance of mathematics and discovery of another physical phenomena, that of the propagation of electro-magnetic waves by the equations attributed to Maxwell (though in reality he is only responsible for one of the 4 equations), provided the first assault on this deterministic world view. The Maxwell equations are described by partial differential equations, whose Hamiltonian formulation is less clear. It is clear now that the pure propagtion of electro-magnetic waves are governed simply by a linear wave equation, and that such equations have uniquely existing solutions given some initial data (for example, the formula of Duhamel gives precisely what to expect at a future time given the initial magnetic and electric field strengths at time 0). But work on that front is still not complete: if the waves are to interact with matter, the system becomes non-linear and predictable solutions are only known for certain types of non-linearities and often only for initial data that are small (i.e. we predict that the universe, if completely empty, will not evolve very much differently from the universe which has a teeny-weeny bit of matter here and an itty-bitty chuck of electromagnetic fields there).

Later in the 20th century, the theory of chaos became known. First in Poincaré's refutation of Laplace's proof that the solar system is stable, and later in studies of continuum mechanics, weather systems, and many more. On the one hand, the idea of chaos theory is that while the system is still integrable: that perfect knowledge of initial conditions afford a perfect prediction of future trajectories, so mathematical determinism is preserved; on the other hand, chaos theory also claims a sensitivity on the initial condition, best summed up in the statement of the butterfly effect. So one interpretation is that given our (humans') limited knowledge of the natural world, either because inaccuracies/inprecision of the measuring instrument, or some innate inability for humans to 'know' the numerical quantity in question to infinitely many decimal places, that the future is, for all practical purposes, not predictable. In particular, that means weather forecasts beyond one week is completely meaningless. Notice that this assault is more on a philosophical level: mathematically the system is still well determined, only that as non-omniscient beings, we humans cannot fully exploit the determinism of the system.

A similar objection to determinism was raised in the middle of the twentieth century when the implications of quantum mechanics become known. Quantum mechanics, in a sense, combined the bad parts of both of the previous two problems. On the one hand, quantum mechanics is described by a partial differential equation (the Schrödinger equation) which, like the wave equations, has a well understood linear theory but a non-linear theory that is extremely difficult. Here we have that the mathematical theory itself does not guarantee uniquely existing solutions for all sorts of interactions possible. On the other hand, quantum mechanics also has the built in "uncertainty principle" which guarantees that observers cannot have perfect knowledge of the position of momentum of a particle, even for the linear problem. (An important philosophical observation that I subscribe to, is that the uncertainty and all that hocus-pocus about observations and Schrödinger's cat are on the level of observations. The actual evolution of the linear problem [i.e. the formula that described the underlying wavefunction] is well defined. It is the evaluation of the wavefunction to produce a tangible results, i.e., an observation, that produces the difficulties.) Regardless of my personal philosphy, the fact remains that on the human level (when we interact with nature by making observations) the theory of quantum mechanics makes predicting the future impossible.

Now, one of the most cited arguments for determinism is the quote of Einstein's that says "God does not play dice." Some people have taken this to mean that Einstein's theory of relativity is a completely deterministic theory. On the contrary, Einstein's theory suffers, on a philosophical level, and on a mathematical level, many of the problems raised before.

Firstly, the problems of mathematics. The theory of special relativity, together with conservation laws (obtained from Noether's theorem), describes a universe not too unlike the Newtonian one. If we consider only Newtonian-style physics: interaction of particles through force, then the theory is similarly integrable, i.e. described by ordinary differential equations that are also subject to chaos. Now if one introduces electromagnetism into special relativity, the same mathematical problems with wave equations still applies. In the theory of general relativity, the problem is only made worse. Here the structure of space-time is also a variable that can evolve, and the evolution equations for space-time happened to be described by a highly non-linear wave equation. Luckily, through the work of Choquet-Bruhat, the wave equation describing an empty universe with no matter what-so-ever has been shown to have uniquely existing solutions (roughly speaking; there are conditions and terms of existence that I am glossing over here) (in general relativity, a universe with no matter can still have interesting structures, for example, there can be propagation of gravitational waves [we're not going to ask what made the waves in the first place]; this is not unlike the fact that in Maxwell theory, electromagnetic waves can be described in a universe that is devoid of any charged objects). The case with matter is less clear. Furthermore, sensitivity to initial condition of the vacuum cases (which Choquet-Bruhat described) is hardly known at all: the only result we have is that of Klainerman and Christodoulou that says that the Minkowski space (the space of special relativity) is a stable solution: i.e., if we started out with an initial empty universe not too different from Minkowski space, we'll continue to have an universe not too different from Minkowski space. But the stability result does not extend to any other configurations. One can think of this as the analogue of the two body problem in celestial mechanics: it is known that when we start we only two celestial bodies in orbit about each other, chaos cannot form. We change the system a little bit (perhaps change the mass of one object a little bit, or its initial direction of motion a little bit, or perhaps add a speck of dust somewhere to the universe), the evolution of the system stays largely the same. But once the problem becomes a three body problem, sensitivity to the initial data can make chaos pop up.

Secondly, with the problem of chaos we start thinking philosophically about initial data. Not only could there be a sensitivity to initial data that destroys stability, whether we, as humans, can have meaningful initial data is doubtful. This goes back to a property of the wave equation known as "finite speed of propagation". For most people, this is also more familiarly known as the principle from special relativity that "nothing can travel faster than the speed of light." As it turns out, for solving the wave equations (or, the Einstein equations in vacuum, a la Choquet-Bruhat), one needs to know the spacial content of the universe at a time slice. Logically this makes sense, as the fact that "nothing can travel faster than the speed of light, not even information" implies that to find the solution to the wave equation at any given point in the future, it suffices to know the information now at all points that can potentially influence that one point we care about. But if we were to predict the evolution of the entire universe for all of eternity, we need to know information now at all points that can potentialy influence the entirety of the universe, so we need to know the information for the entire universe at this precise moment. Now, if we were omniscient gods, this would be trivial. But as we do not have intuitive access to perfect knowledge, we, as humans, make observations. How do we observe, and measure, objects and events in the universe? We look at them. How does visual knowledge come about? It comes about with light interacting with our eyes. So as independent observers, we humans can only ascertain the events of the past that directly influence our being. So now we have a chicken and egg problem: in order to predict the future at some time, we need to know all the information that can contribute to the happenings at that time, but to know all the information that can contribute to the happenings at that time, we need to be at that time already...

So in view of this second problem, how is it that we humans can make any sensible predictions? How is it that we can predict the flight path of a baseball and catch it as it sails over the stadium into the bleachers? Well, the answer is two-fold. Firstly, most things we are familiar with move at speeds rather slower than the speed of light. The things that move rather close to the speed of light usually don't have any direct impact on the tangible events that we humans interact with (when's the last time a neutrino accidentally made the Yankees lose to the Red Sox?). So the information that the baseball, at some time in the past (say, three seconds ago), was flying through the air at the top if its arc with certain speed and that the wind is blowing thus right at that time and a seagull is flying dangerously close to it, reaches our brains much faster than the action of the seagull kicking the baseball a little bit and the ball started careening toward your noggin, so that by the time the ball would have hit your noggin you are not caught by surprise by rather had the time to duck so that the dude sitting behind you actually ended up catching the game-winning homerun. Secondly, and more importantly to us humans, is that the things that do happen at the speed of light doesn't interact much with our life or, if they do, happen so rarely that for all intents and purposes we can assume that they don't happen. Basically, this is saying that the sun will go on shining, the earth will continue to be a oblong sphere, gravity will continue to attract apples falling from trees, and the Yankees will continue to win (okay, maybe not that last one). Human experience (and power of induction) tells us that gravity will continue to pull on the ball as hard as it has been for the past millenia. So even though you could not have known until the time the baseball would've hit your noggin that the gravity of the ball park would not have changed suddenly in the three seconds between your observing the ball changing course and your eventual actions, you can be fairly certain that if you didn't duck the ball would not careen off to outer space or suddenly drop toward second base because it has mysteriously become the center of a new black hole, and that you had better duck if you don't want an icepack on your head 10 minutes later.

So in conclusion: yes, physics and mathematics generally describes a deterministic universe, but that doesn't stop us humans from being stupid and consequently unable to completely predict the future.